Optimal. Leaf size=607 \[ -\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (5 \left (1-\sqrt{3}\right ) b^{2/3} c-2 a^{2/3} e\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} a^{5/3} \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{5 \sqrt{2-\sqrt{3}} \sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2\ 3^{3/4} a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt{a+b x^3}}-\frac{c \sqrt{a+b x^3}}{a^2 x}+\frac{5 \sqrt [3]{b} c \sqrt{a+b x^3}}{3 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 d \sqrt{a+b x^3}}{3 a^2}-\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}} \]
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Rubi [A] time = 0.559812, antiderivative size = 607, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {1829, 1835, 1832, 266, 63, 208, 1886, 261, 1878, 218, 1877} \[ -\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (5 \left (1-\sqrt{3}\right ) b^{2/3} c-2 a^{2/3} e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} a^{5/3} \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{5 \sqrt{2-\sqrt{3}} \sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2\ 3^{3/4} a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt{a+b x^3}}-\frac{c \sqrt{a+b x^3}}{a^2 x}+\frac{5 \sqrt [3]{b} c \sqrt{a+b x^3}}{3 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 d \sqrt{a+b x^3}}{3 a^2}-\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1835
Rule 1832
Rule 266
Rule 63
Rule 208
Rule 1886
Rule 261
Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2}{x^2 \left (a+b x^3\right )^{3/2}} \, dx &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt{a+b x^3}}-\frac{2 \int \frac{-\frac{3 b c}{2}-\frac{3 b d x}{2}-\frac{1}{2} b e x^2-\frac{b^2 c x^3}{2 a}-\frac{3 b^2 d x^4}{2 a}}{x^2 \sqrt{a+b x^3}} \, dx}{3 a b}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt{a+b x^3}}-\frac{c \sqrt{a+b x^3}}{a^2 x}+\frac{\int \frac{3 a b d+a b e x+\frac{5}{2} b^2 c x^2+3 b^2 d x^3}{x \sqrt{a+b x^3}} \, dx}{3 a^2 b}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt{a+b x^3}}-\frac{c \sqrt{a+b x^3}}{a^2 x}+\frac{\int \frac{a b e+\frac{5}{2} b^2 c x+3 b^2 d x^2}{\sqrt{a+b x^3}} \, dx}{3 a^2 b}+\frac{d \int \frac{1}{x \sqrt{a+b x^3}} \, dx}{a}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt{a+b x^3}}-\frac{c \sqrt{a+b x^3}}{a^2 x}+\frac{\int \frac{a b e+\frac{5}{2} b^2 c x}{\sqrt{a+b x^3}} \, dx}{3 a^2 b}+\frac{d \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{3 a}+\frac{(b d) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx}{a^2}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt{a+b x^3}}+\frac{2 d \sqrt{a+b x^3}}{3 a^2}-\frac{c \sqrt{a+b x^3}}{a^2 x}+\frac{\left (5 b^{2/3} c\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{6 a^2}+\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 a b}-\frac{\left (5 \left (1-\sqrt{3}\right ) b^{2/3} c-2 a^{2/3} e\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{6 a^{5/3}}\\ &=\frac{2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt{a+b x^3}}+\frac{2 d \sqrt{a+b x^3}}{3 a^2}-\frac{c \sqrt{a+b x^3}}{a^2 x}+\frac{5 \sqrt [3]{b} c \sqrt{a+b x^3}}{3 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{5 \sqrt{2-\sqrt{3}} \sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{2\ 3^{3/4} a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\sqrt{2+\sqrt{3}} \left (5 \left (1-\sqrt{3}\right ) b^{2/3} c-2 a^{2/3} e\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} a^{5/3} \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.0984323, size = 121, normalized size = 0.2 \[ \frac{-3 c \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{1}{3},\frac{3}{2};\frac{2}{3};-\frac{b x^3}{a}\right )+2 d x \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^3}{a}+1\right )+e x^2 \left (\sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right )+2\right )}{3 a x \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 825, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{3} + a}{\left (e x^{2} + d x + c\right )}}{b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.6304, size = 267, normalized size = 0.44 \begin{align*} d \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{3}}{a}}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} + \frac{a^{3} \log{\left (\frac{b x^{3}}{a} \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} + \frac{a^{2} b x^{3} \log{\left (\frac{b x^{3}}{a} \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} - \frac{2 a^{2} b x^{3} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}}\right ) + \frac{c \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{3}{2} \\ \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} x \Gamma \left (\frac{2}{3}\right )} + \frac{e x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{4}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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